Average Error: 48.0 → 16.6
Time: 12.2s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -9.9024185898810585 \cdot 10^{-13} \lor \neg \left(i \le 0.0066162252799360707\right):\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -9.9024185898810585 \cdot 10^{-13} \lor \neg \left(i \le 0.0066162252799360707\right):\\
\;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\

\end{array}
double f(double i, double n) {
        double r127624 = 100.0;
        double r127625 = 1.0;
        double r127626 = i;
        double r127627 = n;
        double r127628 = r127626 / r127627;
        double r127629 = r127625 + r127628;
        double r127630 = pow(r127629, r127627);
        double r127631 = r127630 - r127625;
        double r127632 = r127631 / r127628;
        double r127633 = r127624 * r127632;
        return r127633;
}
↓
double f(double i, double n) {
        double r127634 = i;
        double r127635 = -9.902418589881058e-13;
        bool r127636 = r127634 <= r127635;
        double r127637 = 0.006616225279936071;
        bool r127638 = r127634 <= r127637;
        double r127639 = !r127638;
        bool r127640 = r127636 || r127639;
        double r127641 = 100.0;
        double r127642 = 1.0;
        double r127643 = n;
        double r127644 = r127634 / r127643;
        double r127645 = r127642 + r127644;
        double r127646 = pow(r127645, r127643);
        double r127647 = r127646 - r127642;
        double r127648 = r127641 * r127647;
        double r127649 = r127648 / r127644;
        double r127650 = r127642 * r127634;
        double r127651 = 0.5;
        double r127652 = 2.0;
        double r127653 = pow(r127634, r127652);
        double r127654 = r127651 * r127653;
        double r127655 = log(r127642);
        double r127656 = r127655 * r127643;
        double r127657 = r127654 + r127656;
        double r127658 = r127650 + r127657;
        double r127659 = r127653 * r127655;
        double r127660 = r127651 * r127659;
        double r127661 = r127658 - r127660;
        double r127662 = r127661 / r127634;
        double r127663 = r127662 * r127643;
        double r127664 = r127641 * r127663;
        double r127665 = r127640 ? r127649 : r127664;
        return r127665;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original48.0
Target47.6
Herbie16.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 2 regimes

  2. if i < -9.902418589881058e-13 or 0.006616225279936071 < i

    1. Initial program 30.1

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
    2. Using strategy rm

    3. Applied associate-*r/30.0

      \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}}\]

    if -9.902418589881058e-13 < i < 0.006616225279936071

    1. Initial program 58.3

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]

    2. Taylor expanded around 0 26.1

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{i}{n}}\]
    3. Using strategy rm

    4. Applied associate-/r/8.9

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -9.9024185898810585 \cdot 10^{-13} \lor \neg \left(i \le 0.0066162252799360707\right):\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 100 (/ (- (exp (* n (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) 1) (/ i n)))

  (* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))))